NeuroImage 54 (2011) S30–S36

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Blast-induced electromagnetic fields in the brain from bone piezoelectricity

Ka Yan Karen Lee a,⁎, Michelle K. Nyein b, David F. Moore c, J.D. Joannopoulos d, Simona Socrate e,Timothy Imholt f, Raul Radovitzky b, Steven G. Johnson g

a Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge MA 02139, USAb Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge MA 02139, USAc Defense and Veterans Brain Injury Center, Walter Reed Army Medical Center, Building 1, Room B207, 6900 Georgia Ave. NW, Washington DC 20309, USAd Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139, USAe Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA 02139, USAf Raytheon Co., 870 Winter St., Waltham MA 02451, USAg Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USA

⁎ Corresponding author.E-mail address: kylkaren@mit.edu (K.Y.K. Lee).

1053-8119/$ – see front matter © 2010 Elsevier Inc. Aldoi:10.1016/j.neuroimage.2010.05.042

a b s t r a c t

a r t i c l e i n f o

Article history:Received 25 January 2010Revised 5 May 2010Accepted 16 May 2010Available online 12 June 2010

In this paper, we show that bone piezoelectricity—a phenomenon in which bone polarizes electrically inresponse to an applied mechanical stress and produces a short-range electric field—may be a source ofintense blast-induced electric fields in the brain, with magnitudes and timescales comparable to fields withknown neurological effects. We compute the induced charge density in the skull from stress data on the skullfrom a finite-element full-head model simulation of a typical IED-scale blast wave incident on anunhelmeted human head as well as a human head protected by a kevlar helmet, and estimate the resultingelectric fields in the brain in both cases to be on the order of 10 V/m in millisecond pulses. These fields aremore than 10 times stronger than the IEEE safety guidelines for controlled environments (IEEE StandardsCoordinating Committee 28, 2002) and comparable in strength and timescale to fields from repetitiveTranscranial Magnetic Stimulation (rTMS) that are designed to induce neurological effects (Wagner et al.,2006a). They can be easily measured by RF antennas, and may provide the means to design a diagnostic toolthat records a quantitative measure of the head's exposure to blast insult.

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Introduction

In this paper,we show that a blast pressurewave traversing the skullis a direct source of potentially intense electric fields in the brain, whichmay have significant neurological effects. This unexpected source doesnot appear to have been considered in studies of primary blast traumaticbrain injury (TBI) (blast-induced neurotrauma (Cernak and Noble-Haeusslein, 2009)). The mechanism is based on the fact that bone is apiezoelectric material: it polarizes electrically in response to an appliedmechanical stress and produces a short-range electric field. Ashockwave from an explosion generates large stresses in the skull and,consequently, large electric fields. Using computed stresses from full-head-model simulations of typical shockwave exposures from impro-vised explosive device (IED) blasts, we calculate the induced chargedensity in the skull and estimate the resulting electric fields. We findfields in the brain on the order of 10 V/m in millisecond-scale pulses,more than 10 times stronger than the corresponding IEEE safetyguidelines for controlled environments (IEEE Standards CoordinatingCommittee 28, 2002), and comparable in strength and timescale to

fields from repetitive transcranial magnetic stimulation (rTMS) that areknown to have neurological effects (Wagner et al., 2006a). Independentofwhether thesefields play a role in TBI, they can be easilymeasured byRF antennas andmay therefore provide a direct measure of the stresseson the skull: a ‘blast dosimeter’ that could be useful for diagnosis andstudy of blast-induced TBI.

In order to assess the potential neurological impact of any blast-induced electromagnetic fields in the brain, we compare them topublished safety standards and also to medical procedures whereneurological effects are intentionally induced in the brain via electricfields. At millisecond timescales (ms pulses, ≈1 kHz frequencies)typical of IED-scale blasts, the IEEE safety standard for in-brain electricfields is 0.3 V/m for the general public and 0.9 V/m in controlledenvironments (IEEE Standards Coordinating Committee 28, 2002).Another point of reference is the medical procedure rTMS, which usesmagnetic-field pulses to create electric fields in the brain that, in turn,induce currents which can disrupt brain activity in the short term orhave long-term effects by stimulating the release of neurochemicals(Wagner et al., 2006b; Muller et al., 2000). A recent full-head finite-element simulation of a typical commercial rTMS device foundmaximum in-brain currents densities of around 4.4 A/m2 (in ms-scale pulses) (Wagner et al., 2006a); the brain has a conductivity ofabout 0.28 A/Vm at kHz frequencies (Wagner et al., 2006a), and hence

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this generates in-brain electric fields of up to 4.4/0.28≈16 V/m. Evenstronger long-term neurological effects, such as retrograde amnesia,are produced by electroconvulsive therapy (ECT), which uses mscurrent pulses repeated several times a second with amplitudes up to1 A (American Psychiatric Association Committee on Electroconvul-sive Therapy, 2002); given the conductivity of the brain and the factthat the ECT current travels over a distance of around 20 cm, a simpledimensional analysis gives an estimated in-brain electric field of (1 A)/(0.28 A/Vm)/(0.2 m)2≈100 V/m. All of these procedures utilizemillisecond pulses because that is the timescale of the neuron actionpotential (Bear et al., 2006), and this unfortunately also means that themillisecond pulses induced by IED-scale explosions are on a timescalethat is neurologically relevant.

An explosion can produce electromagnetic (EM) fields in severalways. First, the initial explosion generates a flash of EM radiation(including visible light), which is measurable far away (Fine, 2001;Kelly, 1993; Boronin et al., 1990; Adushkin and Soloviev, 2004). Inevaluatingfields generatedoutside thebrain, however, it is important torealize that suchfields are reduced in the brain by a factor of roughly therelative permittivity of the brain matter, (Jackson, 1998) which is about105 at the neurologically important kHz frequencies. (Gabriel et al.,1996) Although EM pulses from this initial flash continue to be studiedin many contexts, here we instead consider the generation of EM fieldsfrom the supersonic blast wave itself (outside of the detonation regionwhere the initial flash is generated). For example, the supersonic shockfront is associated with a temperature increase that can ionize particlesin the gas or other materials it passes through, but a simple estimatesuggests that the resultingfields are orders ofmagnitude below the IEEEsafety limits for IED-scale explosions in air (see appendix). Also, a high-pressure blast front can polarizewatermolecules (andother polarizableparticles), an effectwhichhas been observed to yield electricfields up to200 V/m for very high pressures (100 kbar) corresponding to nuclear-scale explosions, (Harris, 1965; Harris and Presles, 1982; Allison, 1965;Hauver, 1965; Linde et al., 1966; Hayes, 1967; Frankel and Toton, 1979)but this effect should be negligible for IED-scale blasts where thepressures aremuch lower (0.02 kbar (Moore et al., 2009)). Finally,muchlarger EM fields can be generated when the high-pressure blast waveimpacts a piezoelectric material (Reed et al., 2008; Shkuratov et al.,2004), which polarizes even in response to low pressures (and aretherefore used as pressure sensors and actuators for a variety ofapplications (Crawley and de Luis, 1987; Anderson and Hagood, 1994;Near, 1996; Shkuratov et al., 2004)). Here, the key fact is that bone isknown to be a strong piezoelectric material (Fukada and Yasuda, 1957;Bassett and Becker, 1962; Cochran et al., 1989; Black and Korostoff,1974; Reinish and Nowick; Pfeiffer, 1977; Bur, 1976; Singh and Katz,1988; Williams and Breger, 1975; Halperin et al., 2004; Aschero et al.,1996): even though a polarized piezoelectric material is neutral (no netcharge) and the resulting fields are short range, the adjacency of skull

Fig. 1. Schematic generation of charges from a blast by bone piezoelectricity: (left) an incomiskull (middle: skull pressure). This causes the bone to polarize and leads to an electric chargregions of large positive (red) and negative (blue) charge density are created which lead to

bone to the cerebral cortex means that even short-range fields may berelevant to TBI if they are strong enough.

The piezoelectric effect of bone has been observed since Fukada(Fukada and Yasuda, 1957), and was originally thought to play a role instimulating bone growth (Marino and Becker, 1970). In response to ashockwave from an explosion, electric fields are produced in the brainvia the process depicted schematically in Fig. 1. First, an incoming blastwave in the air hits the soldier's head from the side. Stresses are inducedin the tissues in thehead. Then, an electrical chargedensity is createdviathe bone piezoelectric effect due to the tensile and shear stresses. Thetime scale of these fields is directly related to the timescale of the shock(as described in more detail below, the field is a linear function of theapplied stresses). Inparticular, a typical IED explosion has amillisecond-scale high-pressure duration, which results in a ms-scale electric fieldpulse, corresponding to kilohertz-scale (kHz) frequencies f (from theFourier transform of a ms pulse), at which thewavelength c/ f of light ishundreds of kilometers in air (and at least hundreds of meters even inbrainmatterwhere the refractive index is 103 (Gabriel et al., 1996)). Thismeans that neither the wave nature of the light nor the inducedmagnetic fields are relevant on the scale of a human head: theelectromagnetic response is accurately described by an electrostaticmodel in which purely electric fields are produced instantaneously inresponse to the charge density (Jackson, 1998).

The remaining paper is organized as follows. In Methods section,we describe the methodology of our work. We first review thephenomenon of bone piezoelectricity in Bone piezoelectric responsesection, summarize the available measurement data, and describe thevalues used in this paper. Then we outline howwe calculate the blast-induced charge densities (Blast-induced charge density) and electricfield estimates (Electric field estimation). Results section presents theresults we obtain: charge densities and estimated electric fields as afunction of time and space for an IED-scale blast, as well as aninvestigation of the sensitivity of the results to the precise value of thebone's piezoelectric response. Finally, we conclude with futuredirections and some implications of our results.

Methods

Bone piezoelectric response

Piezoelectricity is a phenomenon in which an electric polarizationis generated in response to an applied mechanical stress:

P = dσ ð1Þ

where P, d, and σ are the polarization density, the piezoelectric tensor(a property of thematerial), and the stress tensor, respectively (Brainerdet al., 1949). It iswell known that the piezoelectric effect occurs in certain

ng blast wave in the air impacts a soldier's head from the left. This generates stress in thee density (right) proportional to the stress (and its gradient). The net charge is zero, butshort-range electric fields.

Fig. 2. Average and maximum piezoelectric charge density magnitude |ρ| as a functionof time, as well as the corresponding peak electric field estimate, over the course of atypical IED-scale blast impact for the unhelmeted human head.

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classes of crystals and ceramics. Numerous efforts to measure suchproperties in biomaterials have been undertaken. In particular, FukadaandYasudademonstrated apiezoelectric response indrybone at 2 kHz in1957 and measured a typical piezoelectric coefficient d14 (describedbelow) to be 0.2 pC/N (Fukada and Yasuda, 1957) (about one tenth of thevalue for quartz). Subsequent measurements have shown that piezo-electricity is also present in wet bone (Bassett and Becker, 1962), livebone (Cochran et al., 1989; Black and Korostoff, 1974), and in otherbiomaterials such as tendons (Williams and Breger, 1975), collagen(Fukada et al., 1976), skin (Shamos and Lavine, 1967), and blood-vesselwalls (Fukada and Hara, 1969). Reinish and Nowick established thatmoisture level plays a role in bone piezoelectricity (Reinish andNowick ),and Pfeiffer studied the effect of the frequency of the applied stress onbone piezoelectricity (i.e., material dispersion) (Pfeiffer, 1977). Bur(1976) studied bonepiezoelectric properties as a functionof temperatureandhumidity in the range10−2 to102 Hz.Anextensive summaryof thesemeasurements can be found in Singh and Katz (1988), and more recentmeasurements were described in Halperin et al., (2004) and Ascheroet al., (1996). These values were all measured for animal or human tibiasand femurs, and the measured piezoelectric coefficients have somevariability (within the same order of magnitude). However, the largestcoefficients are all on the order of 0.1 pC/N at kHz frequencies. In thiswork, we have chosen to take into account only bone piezoelectricitybecause the stresses in the bones are much higher than those of otherbiological materials in and around the brain (Moore et al., 2009), and atthis point we are mainly interested in order-of-magnitude estimates.

The piezoelectric coefficient d is a rank-3 tensor (a “3d matrix”)encoding the polarization in response to different types andorientations of the stress. As we show in Results section, the preciseorientation and anisotropy of d appear to make little difference in theresulting charge density in the skull from a blast wave (nomore than afactor of 4) as long as the magnitude of the components of d is thesame (technically, fixed Frobenius norm ∥d∥F). As long as theorientation does not vary too rapidly in space (since rapid oscillationsin d would yield canceling polarizations), the orientation is thereforenot crucial in estimating the order of magnitude of the resultingelectric field. Nevertheless, we consolidate the available experimentaldata to obtain a realistic piezoelectric tensor:

d =0 0 0 10:5 −1:3 00 0 0 −1:3 −10:5 03:8 3:8 0:5 0 0 0

24

35 × 10−14C =N ð2Þ

where axes 1 and 2 are tangential to the plane of the skull and axis 3 isnormal to the plane of the skull. Thus, these axes rotate relative to thelocation on the skull. For simplicity, we assume that the piezoelectricresponse in the skull is rotationally invariant around axis 3. (Here, weuse the standard encoding of the rank-3 tensor d as a matrix (Brainerdet al., 1949), where the six columns correspond to the six degrees offreedom in the symmetric stress tensor σ.) This piezoelectric tensorpossesses rotational symmetry in the tangent plane of the skull. For thecoefficients, we used values typical of those reported in a variety ofmeasurements (Bur, 1976;Aschero et al., 1996),whered14=0.105pC/Nis the largest coefficient as reported by Ref. Bur. The d14 coefficient infemur corresponds tobending thebonealong its length, and sowe chosethe corresponding coordinate system for the skull so that d14corresponds to bending the skull, andwe assumed rotational invariancewith respect tobending aroundany axis in the tangent planeof the skull.Again, we will see that the precise arrangement of the coefficients in dseems not to matter very much for the peak order of magnitude of theresulting in-brain electric field.

Blast-induced charge density

We calculate the charge density using stress data on the skull froma finite-element full-head-model simulation (Moore et al., 2009) of a

typical IED-scale blast wave incident on an unhelmeted human headfrom the side, as well as on a human head protected by a kevlarhelmet (Advanced Combat Helmet).

As described inMoore et al., 2009, these simulations employeda full-head tetrahedral mesh in the context of a blast with an overpressure of30 atm (0.03 kbar): the 99% lethal dose (LD99) for lung-injury survival inan unarmored person (albeit survivable with current personal-protection equipment), equivalent to 0.569 kg of TNT at a 0.6 mstandoff distance. (As described in Ref. (Moore et al., 2009), the headmodel was generated with 1 mm3 resolution frommagnetic-resonanceimages (Collins et al., 1998) combinedwith a bone-windowed CT scan.)The computational head model differentiates 11 distinct biologicalmaterials characterized by different mechanical properties such asnonlinear viscoelasticity, anisotropy and strong strain rate dependence.The simulationconsists offluid–solid interactionwitha blast shockwaveunder open boundary conditions.

To obtain the polarization by the piezoelectric effect from thestresses, wemultiply d by σ according to Eq. (1). (Since the coordinatesystem of σ is fixed in space, while the coordinate system of d is skull-relative, we first rotate σ into the skull-relative coordinate system foreach location on the skull, and then rotate back the resulting P.) Thisgives us the polarization vector P (in the fixed coordinate system of σ)at each vertex of the finite-element model. The electric polarizationproduces a local charge density via the following formula (Jackson,1998):

ρ = −∇⋅P ð3Þ

where ρ is the charge density per unit volume. We compute ∇·P bylinear interpolation of the polarization between the vertices of eachtetrahedron (corresponding to first-order elements for P) to obtainthe charge density on a dual mesh located at the center of eachtetrahedron. Note that the combination of Eq. (1) and Eq. (3) meanthat not only large stresses but also large stress gradients give rise tolarge charge density ρ, which leads in Results section to large chargedensities around small features such as eye sockets where there arelarge stress variations.

Electric field estimation

In many regions, we find (below) that the charge density in theskull locally resembles a sheet of charge. This allows us to estimate thefield near the skull in these regions, because the field of a sheet ofcharge is known analytically, an approximation that should be validclose to the skull where the field is most intense. Therefore, we use

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this as a first approximation to estimate the maximum field mag-nitudes in the brain.

In particular, electric field very close to a sheet of charge is insensitiveto the finite size of the charged region, and therefore can beapproximated by the known field of an infinite charged plane. In auniform dielectric, that field has amagnitude of ρs/2ε, where ρs and ε arethe surface charge density and the permittivity of the material,respectively (Jackson, 1998). For a surface charge density at the interfacebetween twomaterialswithpermittivities ε1 and ε2, it canbe shown fromthe method of images (Jackson, 1998) that the corresponding fieldmagnitude in both materials is ρs/2εeff where εeff=(ε1+ε2)/2 is theaverage permittivity. For the particular case of a volume charge density ρ

Fig. 3. Computed charge density distributions in the skull region from front view (top left) ablast has an overpressure of 30 atm as it impacts the head, corresponding to 0.569 kg of TNTcase, and 0.347 ms for the helmeted case.

in a thin sheet of thickness h, the corresponding surface charge density isρs=ρh. Therefore, we obtain the following estimate for the electric fieldmagnitude near a charged region of the skull:

jE j≈ ρh2εeff

ð4Þ

≈ρh= 2ε01 + 105

2

!ð5Þ

≈ρ⋅2300Vm2= C ð6Þ

nd side view (middle left), along with the various labeled cross sections. The associatedat a distance of 0.6 m. The time is 0.548 ms after impacting the head for the unhelmeted

Fig. 4. Histogram of maximum charge densities calculated for 100 randomly orientedpiezoelectric tensors. (The tensor components were chosen as uniform randomnumbers in [−0.5,+0.5], and then d was rescaled to have a fixed Frobenius norm∥d∥F = 0:159 × 10−12 C/N.) The results show that the precise details of thepiezoelectric orientation in cranial bone are unlikely to change the order of magnitudeof the peak charge density, as long as the strength of the piezoelectric effect is the same.

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where ρ is the charge density, h=0.002m is the thickness of the skull,ε0=8.85⋅10−12 F/m is the vacuum permittivity, and the relativepermittivity ε /ε0 of air and brain matter are 1 and roughly 105 (at kHzfrequencies (Gabriel et al., 1996)), respectively.

Results

Using the methods described in the preceding section, weobtained the charge density in the skull and the resulting electricfield magnitude estimates in the brain regions near the skull. In thissection, we present these results, starting with the time and spacedependence of the charge density and the strength of the resultingfields, and concluding by investigating the sensitivity of these resultsto the precise details of the piezoelectric coefficient d.

Fig. 2 displays the average and maximum charge densitymagnitude |ρ| in the skull bone, as well as the corresponding electricfield estimates with respect to time over the course of the blast waveimpact. An immediate observation is that ρave≪ρmax, with theaverage |ρ| being more than an order of magnitude smaller than themaximum |ρ|. This is consistent with the observation from Fig. 3 thathigh charge densities are often concentrated in small (cm-scale)regions. For example, we can see higher charge densities in the skullon the side facing the shockwave impact, both in the center of the sideand also around the eye sockets where there are small features in theskull leading to large stress gradients (as mentioned in Blast-inducedcharge density section). The corresponding estimated maximumelectric field from Eq. (4) is 6 V/m. The peak charge density andelectric field are changed by less than 20% in the case of a helmetedhead: a typical charge density for the helmeted case is shown in Fig. 3(f), and displays similar peak densities but over a smaller area of thecranium. This maximum field is attained in a millisecond-scale pulse(corresponding to the duration of the high-pressure blast front), andso should be compared to safety standards andmedical procedures formillisecond-scale pulses (equivalent to the standards for kHzfrequencies (IEEE Standards Coordinating Committee 28, 2002)). Inparticular, our estimated field magnitude on the order of 10 V/m areabout an order of magnitude larger than the IEEE safety standards andare comparable to the electric fields used in rTMS, as reviewed in theIntroduction.

Since the piezoelectric coefficient d has not been measured forhuman or animal skulls, and there is some variation in measured dataeven for tibias and femurs, we investigate how sensitive the chargedensities and resulting electric fields are to the precise details ofd. Here,we assume that the orientation of the piezoelectric tensor is coherent(that the tensor orientation changes on the scale of at least centimeters,the lengthscale of coherence in our stresses andchargedistributions), anassumption that has yet to be checked experimentally for the craniumbut is consistent withmeasurements for femurs and tibias (which havea specific d orientation relative to the long axis of the bone). If d wererotating rapidly across the skull, then the polarizations would bepointing in different directions and could mostly cancel.

We proceed with our sensitivity analysis. First, the charge densityand electricfield are clearly linear functions ofd fromEqs. (1) and (3), sosimply multiplying d by a constant will scale ρ and E by the sameconstant. The only remaining question is the sensitivitywith respect the“orientation” of d: the distribution of the components of the d tensorwith a fixed “average” magnitude. We quantify this by fixing theFrobenius norm ∥d∥F (the root-mean-square value of the componentsof d), imposing rotational symmetry in the tangent plane of the skull,and randomly varying the distribution of the components.We generate100 random d tensors with this rotational symmetry, using uniformdistribution to in [−0.5,0.5] assign values to the non-zero degrees offreedom in the tensor and then rescaling to keep the norm ∥d∥F fixed tothat of the tensor in Eq. (2), and using these to calculate the chargedensities as in Blast-induced charge density section. We show in Fig. 4the histogram of the maximum charge density |ρ|, and find that the

maximum |ρ| varies by around a factor of 2–3 from the |ρ| produced byour original value of d. Therefore, it seems that the precise details of thedorientationdonot affect the order ofmagnitudeof the chargedensity orthe resulting in-brain electric fields.

Conclusions

From our investigation, the biggest EM fields in the brain maycome from an unexpected source – piezoelectricity of bone – that hadnot yet been studied in the context of blast TBI. These electric fieldsare estimated to be on the order of 10 V/m over millisecond-durationpulses, exceeding IEEE safety standards by an order of magnitude andcomparable rTMS procedures known to have neurological effects. Thisraises the possibility that piezoelectric fields produced by the blastwave impacting the skull may contribute to blast-induced TBI, apossibility that we believe merits further study. Future work shouldalso consider the possibility that the effect of the electric fields on thebrain is exacerbated by the mechanical blast injury, especially sinceincreased intracranial pressure is already known to increase risks inrTMS (Wassermann, 1998).

There are a number of questions that should be addressed by futureresearch. First,weplan toperformdirect experimentalmeasurementsofshocked cranial bone in order to observe the resulting electric fields,determine the relevant piezoelectric coefficient, and compare withtheoretical predictions. (Future experimental measurements of thepiezoelectric effect in cranial bone should also determine any spatialvariation of the piezoelectric coefficient d across the thickness of thebone or in different regions of the skull; for example, any rapid variationin the magnitude or orientation of the piezoelectric tensor d acrosssutures between cranial boneswould lead togreatly increasedρ at thesejoints.) Second, we are currently incorporating the computed chargedensity into a full three-dimensionalfinite-element solutionof Poisson'sequation −∇⋅(ε∇)ϕ=ρ to find the full three-dimensional fieldsE = −∇ϕ, which will differ from the estimated fields above in regionswhere the charge density is varying rapidly in the skull. Eventually,electromagnetic solvers may need to be directly coupled withshockwave simulations. Ultimately, EM effects may need to be includedin animal models and other research on blast-related TBI.

Regardless of whether piezo-induced electric fields play a role inblast-induced TBI, theymay be useful in research and diagnosis of blast-related injuries. In particular, the samemechanism that produces fields

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inside the brain also generates short-range fields of similar magnitudesimmediately outside the head, and such kHz-frequency fields are easilymeasured by radio-frequency (RF) antennas. Small RF antennasincorporated into protective headgear could therefore provide a formof “blast dosimeter:” if the field strengths are recorded, they can beanalyzed after the blast to provide a directmeasure of the blast-inducedstresses on the skull as a quantitative diagnostic metric [cite patentapplication].

Conflict of interest statement

The authors declare that there are no conflicts of interest.

Acknowledgments

This work was supported in part by the Joint Improvised ExplosiveDevice Defeat Organization (JIEDDO) and the Institute for SoldierNanotechnologies (ISN) through the Army Research Office. We aregrateful to Evan Reed for many helpful suggestions.

Appendix. Ionization of air by IED blast waves

In order to eliminate the possibility that ionization of air by theblast wave can itself lead to neurologically significant electric fields,we present a conservative order-of-magnitude estimate on thermo-dynamic grounds (suggested by E. Reed (Reed, 2008)). Here, weconsider the ionization by the blast wave in air only, not the ionizationwithin the detonation region (the initial flash). It turns out that theinduced electric field in an air blast wave can be characterized interms of the blast wave pressure, temperature, and air-ionizationenergy alone, from thermodynamic considerations, independent ofhow the blast wave was generated, and we use these parameters toestimate the fields from an IED-scale blast below.

First, we estimate the electricfield in terms of the charge density andtemperature. An electric field from ionization must be created by theseparation of charges, and this separation requires energy qVwhere q isthe value of an individual charge (e.g. an electron) and V is the voltage.This energy, if it is supplied by the blast wave, should be comparable tothe thermal energy kT where k is Boltzmann's constant and T istemperature (Reif, 1965), assuming the charges equilibriate on a shortertimescale than the blast wave duration (milliseconds). If the blast wavehas spatial width ℓ and the charge density is ρ, then the peak electricfield will be E∼ρℓ/ε0 (similar to Eq. (4)) and the potential will beV∼Eℓ∼ρℓ2/ε0. Combining thiswith qV∼kT and solving forℓ,weobtainℓ∼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikTε0 = qρ

pand peak fields E∼ρℓ= ε0∼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiρkT = qε0

p.

Second, we obtain the charge density from the Boltzmanndistribution: if the ionization energy of the gas is I and the numberdensity is n, then one should have ρ=qne− I / kT at equilibrium (Reif,1965). This can now be combined with the ideal gas law P=nkTrelating pressure P to n and T, and substituted into the above equationfor E to obtain:

E∼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe−I =kT

ε0

s: ð7Þ

We now make a conservative estimate for E. Even if the tem-peratures in the blast wave were to reach T=1000 K (0.086 eV) withpressures reaching P=100 atm (107 Pa), with the ionization energy ofair being I≈10 eV (slightly smaller than the typical value for gases(Martin and Wiese, 1996)), this equation predicts electric fields ofonly ∼10−16 V/m. Recall, also, that any field produced in air isreduced in the brain by roughly a factor of the relative permittivity(Jackson, 1998) of the brain tissue (∼105 (Gabriel et al., 1996)).

(For nuclear-scale blast waves with even higher temperatures inthe thousands of K, the fields from ionization quickly become more

significant, thanks to the exponential dependence in the Boltzmannfactor e− I / kT, and have been studied in more detail in that context(Glasstone and Dolan, 1977).)

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